These directional spin waves can mediate a chiral interaction between two wires, approached by the Langevin equation under Hamiltonian Eq. (REF ). When the magnetic quality of the wire is higher than that of the film, we are allowed to use the Markov
approximation [1]}, [2]} when integrating out the film degree of freedom, yielding the Langevin equation for wires
\(\frac{d\hat{b}_{l}}{dt} =-i\omega _{\mathrm {K}}\hat{b}_{l}-\frac{\kappa }{2}\hat{b}_{l}-G_{l}(\omega )\hat{b}_{l}-\sum _{l\ne l^{\prime }}G_{ll^{\prime }}(\omega )\hat{b}_{l^{\prime }.}\) 
